Hecke Operators

Abstract

Hecke operators are averaging operators similar to a trace. They operate on the space of modular forms. Let /be a modular form, f = ∑a _n qⁿ, with associated Dirichlet series

$$ \varphi = \sum {{a_n}{n^{ - s}}.} $$

. It turns out that f is an eigenfunction for all Hecke operators if and only if the Dirichlet series has an Euler product. Such Euler products give relations among the coefficients, which show that they are multiplicative in n (i.e. a _mn =a _m a _n if m, n are relatively prime), and that they satisfy certain recurrence relation for prime power indices. The reader will find applications for these in Chapter VI, § 3. One of the basic problems of the theory is to organize into a coherent role the relations satisfied by these coefficients, and their effect on the arithmetic of number fields. The Hecke ones are in a sense the oldest. Later chapters touch on congruence relations. Manin [Man 4] found some which are much more hidden. The situation is very much in flux as this book is written.